On the effective reducibility of a class of Quasi-periodic nonlinear systems near the equilibrium. Issue 3 (3rd July 2019) - Li, Jia; Zhu, Chunpeng
ABSTRACT: In this paper, we consider the effective reducibility of the following quasi-periodic nonlinear system x ˙ = ( A + ε Q ( t, ε ) ) x + ε g ( t, ε ) + h ( x, t, ε ), | ε | ≤ ε 0, where A is a constant matrix with the different and nonzero eigenvalues, h = O ( x 2 ) ( x → 0 ), and h ( x, t ), Q ( t ) and g ( t ) are analytic quasi-periodic on D ρ with respect to t . Under non-resonance conditions, without any non-degeneracy condition, by a quasi-periodic transformation, the system can be reducible to a quasi-periodic system y ˙ = ( A ∗ ( ε ) + ε R ∗ ( t, ε ) ) y + ε g ∗ ( t, ε ) + O ( y 2 ), | ε | ≤ ε 0, where R ∗ and g ∗ are exponentially small in ϵ .